a)The mean of X, is given by:
E(X) = [tex]M'_X(0)=\frac{7}{3}[/tex]
b) The variance of X is :5/9
c) From the properties of the moment generating function of a discrete random variable, the distribution of X is given by:
P(x) = 1/6, x =1
p(x) = 2/6, x = 2
p(x) = 3/6, x = 3
Let:
[tex]M_X(t)=\frac{1}{6}e^t+\frac{2}{6}e^2^t+\frac{3}{6}e^3^t[/tex]
be the moment generating function variable X. Then
[tex]M'_X(t)=\frac{1}{6}e^t+\frac{4}{6}e^2^t+\frac{9}{6}e^3^t\\\\M"_X(t)=\frac{1}{6}e^t+\frac{8}{6}e^2^t+\frac{27}{6}e^3^t[/tex]
[tex]M'_X(0)=\frac{1}{6}e^0+\frac{4}{6}e^2^(^0^)+\frac{9}{6}e^3^(^0^)=\frac{1}{6}+\frac{4}{6}+\frac{9}{6}=\frac{7}{3} \\\\M"_X(0)=\frac{1}{6}e^0+\frac{8}{6}e^2^(^0^)+\frac{27}{6}e^3^(^0^)=\frac{1}{6}+\frac{8}{6}+\frac{27}{6} =6[/tex]
a) The mean of X, is given by:
E(X) = [tex]M'_X(0)=\frac{7}{3}[/tex]
b)The variance of X is given by:
Var(x) = M"(X)(0) - [M'x(0)]^2
= 6 - [tex](\frac{7}{3} )^2[/tex]
= 5/9
(c) From the properties of the moment generating function of a discrete random variable, the distribution of X is given by:
P(x) = 1/6, x =1
p(x) = 2/6, x = 2
p(x) = 3/6, x = 3
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Let : = (a + ai)(b + b/3i) where a and b are positive real numbers. Without using a calculator, determine arg 2.
The answer is arg 2 = tan^-1(3/2) where : = (a + ai)(b + b/3i) where a and b are positive real numbers.
To determine arg 2, we need to first find the value of :.
Expanding the given expression, we get:
: = (a + ai)(b + b/3i)
: = ab + ab/3i^2 + abi + ab/3i
: = ab - ab/3 + abi + ab/3i
: = (2ab/3) + (ab)i
Now, we can find the modulus of : as:
|:| = sqrt((2ab/3)^2 + (ab)^2)
|:| = sqrt(4a^2b^2/9 + a^2b^2)
|:| = sqrt(13a^2b^2/9)
And, we can find the argument of : as:
arg(:) = tan^-1((ab)/(2ab/3))
arg(:) = tan^-1(3/2)
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ANSWER ASAP! DUE TODAY
5. Find the measure of arc AC.
Find mAC
A) 52
B) 72
C) 92
D) 112
Answer:
A. 52°
Step-by-step explanation:
The measure of an inscribed angle is 1/2 the measure of the intercepted arc.
m∠B = 1/2 (m of arc AC)
26 = 1/2 (m of arc AC)
52 = (m of arc AC)
Which groups of numbers could 1/3 fall under. Natural,whole,integer,or rational
Answer:
The number 1/3 is a rational number. A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
1/3 is not a natural number, whole number, or integer. Natural numbers are the set of positive integers (1, 2, 3, …). Whole numbers are the set of non-negative integers (0, 1, 2, 3, …). Integers are the set of whole numbers and their additive inverses (-3, -2, -1, 0, 1, 2, 3, …).
Step-by-step explanation:
whats 40/8 but not 5? if u get it right, u get branliest cuz u big brain
Answer:
it could also be [tex]\frac{20}{4}[/tex] or [tex]\frac{10}{2}[/tex]
Step-by-step explanation:
Simplify.
[tex]\frac{40}{8}[/tex] ÷ 2 = [tex]\frac{20}{4}[/tex]
[tex]\frac{20}{4}[/tex] ÷ 2 = [tex]\frac{10}{2}[/tex]
The aggregate claim amount is defined as S X₁, where Xį are independent and i=1 identically distributed claim amounts and N is the number of claims. (a) Suppose N follows a negative-binomial distribution with E(N) = k(1 − p)/p and variance Var(N) = k(1 − p)/p². Show that: k(1 − p) (i) E(S) = -m₁ and р k(1 − p) k(1-P)² m², (ii) Var (S) = -M2 + p² - (iii) Skew k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P
The required results are:
(i) E(S) = -m₁
(ii) [tex]Var(S) = k(1 − p)(m₂ - m₁²)/p + [k(1-p)/p]² (m₁² + m₂ - m₁²)[/tex]
(iii) Skew = 3(1 − 2p)/sqrt{k(1 − p)}.
The aggregate claim amount is defined as S X₁, where Xį are independent and i=1 identically distributed claim amounts and N is the number of claims.
And also suppose N follows a negative-binomial distribution with
E(N) = k(1 − p)/p
and variance
Var(N) = k(1 − p)/p².
Now we have to show that:
k(1 − p)
(i) [tex]E(S) = -m₁ and р k(1 − p) k(1-P)² m²,[/tex]
(ii) Var (S) = -M2 + p² -
(iii) [tex]Skew k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P(i) E(S) = -m₁[/tex]
Where S = X1 + X2 + ...... + XN.
Also, N and Xi's are independent.
So, E(S) = E(E(S|N))
From the law of total expectation,
E(S|N) = E(X1 + X2 +......+XN|N)
Using linearity of expectation
,E(X1 + X2 +......+XN|N) = E(X1|N) + E(X2|N) +......+ E(XN|N) = NE(X1)Now, E(S) = E(E(S|N)) = E(NE(X1))= NE(X1) = m₁
Therefore,
E(S) = -m₁.(ii) Var (S) = -M2 + p² -Where Var(S|N) = Var(X1 + X2 +......+XN|N) = Var(X1|N) + Var(X2|N) +......+ Var(XN|N)Also, E(S|N) = NE(X1)Thus, E(S) = E(E(S|N)) = E(NE(X1)) = NE(X1) = m₁ And Var(S) = E(Var(S|N)) + Var(E(S|N))
Using independence,
[tex]Var(S|N) = Var(X1 + X2 +......+XN|N) = Var(X1|N) + Var(X2|N) +......+ Var(XN|N)Var(S|N) = N Var(X1)Using independence, E(S|N) = NE(X1)Var(E(S|N)) = Var(NE(X1)) = N²Var(X1)Hence, Var(S) = E(Var(S|N)) + Var(E(S|N))= E(N Var(X1)) + Var(NE(X1))= (k(1 − p)/p) Var(X1) + N²Var(X1)= k(1 − p)Var(X1)/p + N²Var(X1)[/tex]
Also, [tex]Var(X1) = E(X1²) - (E(X1))² = m₂ - m₁²[/tex]
Therefore,
Var(S) = k(1 − p)(m₂ - m₁²)/p + N²(m₂ - m₁²)
And N follows a negative-binomial distribution with
E(N) = k(1 − p)/p and variance
[tex]Var(N) = k(1 − p)/p². Thus, E(N²) = Var(N) + (E(N))²E(N²) = k(1-p)/p² + [k(1-p)/p]²[/tex]
Therefore,
[tex]Var(S) = k(1 − p)(m₂ - m₁²)/p + [k(1-p)/p]² (m₁² + m₂ - m₁²)(iii) Skew = 3(1 − 2p)/sqrt{k(1 − p)}[/tex]
Therefore,
[tex]k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P = 3(1 − 2p)/sqrt{k(1 − p)}.[/tex]To know more about binomial distribution, visit:
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Given that a^b=x, evaluate the following: a^26 PLS HELP NOW
Answer:
Step-by-step explanation:
Find the log of both the equation and the expression:
b log a = log x this can be solved for log a in terms of log x:
log x
log a = ------------
b
"Name" the second expression "y." Then y = a^26 and log y = 26 log a.
log x
Then log y = 26--------------
b
and the expression a^26 is found by taking the antilog of both sides of log y:
log y log x
10 = y = 10^(26----------)
b
Please double check to ensure that you have copied this problem down correctly.
5potatos + 56 potatos
Answer:
61
Step-by-step explanation:
Fill in the blanks to complete the area model to solve 6.60 divided by 15 is the same as _ x ? = _ 6.60 = _ hundredths
Answer:
[tex]15 x = 6.60[/tex]
[tex]x = 44\ hundredths[/tex]
Step-by-step explanation:
Given
[tex]\frac{6.60}{15}[/tex]
Required
Complete the model
[ ] x = 6.60
x = _ hundredths
[tex]\frac{6.60}{15}[/tex]
Equate to x
[tex]x = \frac{6.60}{15}[/tex]
Multiply both sides by 15
[tex]15 * x = \frac{6.60}{15} * 15[/tex]
[tex]15 * x = 6.60[/tex]
[tex]15 x = 6.60[/tex]
So, we have:
[ ] x = 6.60 =====> [tex]15 x = 6.60[/tex]
Recall that:
[tex]x = \frac{6.60}{15}[/tex]
[tex]x = 0.44[/tex]
This implies that:
[tex]x = 44\ hundredths[/tex]
Hence:
x = _ hundredths =====> [tex]x = 44\ hundredths[/tex]
QUESTION 45
What is the value of x?
3x
Answer:
1/67
Step-by-step explanation:
Complete question
If X+2/3x is equal to 45. find the value of x
Given the equation
x+2/3x = 45
Cross multiply
x+2 = 3x(45)
x+2 = 135x
x - 135x = -2
-134x = -2
x = 2/134
x = 1/67
Hence the value of x is 1/67
Each of the following sentences has three errors in grammar, capitalization, usage, or spelling. Write a correct version for each sentence. Avoid adding new phrases, starting new sentences, or rewriting in your own words.
1) Among the oddly named towns in the United States are: What Cheer, Iowa, Peculiar, Missouri, and Cheesecake, New Jersey.
2) After our supervisor and her returned from their meeting at 2:00 p.m., we were able to sort the customers names more quickly.
3) 6 of the 18 workers in my department were fired, as a result we had to work harder to achieve our goals.
4) Jeffreys presentation to a nonprofit group netted him only three hundred dollars, which is a tenth of his usual honorarium but he believes in pro bono work.
5) Of all the discoveries and inventions in human history the 4 greatest are said to be these speech, fire, agriculture, and the wheel.
6) Our latest press release written by our corporate communication department announces the opening of three Asian offices.
7) In his justification report dated September first, Justin argued that expansion to twelve branch offices could boost annual revenue to 22 million dollars.
8) The practicality and advisability of opening 12 branch offices are what will be discussed in the consultants feasibility report.
9) 3 categories of Apps (social media, productivity, and entertainment—are most popular for installing on smartphones.
10) Because some organizations prefer single spaced reports be sure to check with your organization to learn it’s preference.
Below are the corrected versions of the sentences with grammar, capitalization, usage, and spelling errors addressed.
Oddly named towns in the United States include What Cheer, Iowa; Peculiar, Missouri; and Cheesecake, New Jersey.After our supervisor and she returned from their meeting at 2:00 p.m., we were able to sort the customers' names more quickly.Six of the eighteen workers in my department were fired. As a result, we had to work harder to achieve our goals.Jeffrey's presentation to a nonprofit group netted him only three hundred dollars, which is a tenth of his usual honorarium. But he believes in pro bono work.Of all the discoveries and inventions in human history, the four greatest are said to be speech, fire, agriculture, and the wheel.Our corporate communication department has written the latest press release, which announces the opening of three Asian offices.In his justification report dated September 1st, Justin argued that expanding to twelve branch offices could boost annual revenue to 22 million dollars.The feasibility report from the consultants will discuss the practicality and advisability of opening twelve branch offices.Three categories of apps (social media, productivity, and entertainment) are most popular for installing on smartphones.Because some organizations prefer single-spaced reports, be sure to check with your organization to learn its preference.The sentence lists the oddly named towns correctly, but the capitalization of "Iowa" and "New Jersey" should be in lowercase, and colons are not needed in this context.The pronoun "her" should be replaced with "she" to be grammatically correct. "Customers names" should be "customers' names" to show possession.The numbers "6" and "18" should be written as words, and the sentence should be divided into two sentences for clarity."Jeffreys" should be corrected to "Jeffrey's" to indicate possession. "Pro bono work" should be in italics or quotation marks for correct formatting.The sentence lacks proper punctuation and capitalization. The four greatest discoveries and inventions should be separated by commas.The sentence is grammatically correct, but the use of "our" twice can be avoided for better flow.The date should be written as "September 1st" instead of "September first" for proper formatting.The subject-verb agreement is incorrect. "The practicality and advisability" should be followed by "is," not "are."The number "3" should be written as "Three." The parentheses should be placed correctly, and a comma is needed after "entertainment.""It's" should be corrected to "its" as it denotes possession, not a contraction. The sentence should be divided into two sentences for clarity.Learn more about spelling errors here:
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Plz help me we are almost done
Answer:
1) D
2) B
Step-by-step explanation:
Hope this Helps!
:)
log(x)-log(x+6)=log(k)
Find x if k=1/2
Answer:
x = 6
Step-by-step explanation:
Using the rules of logarithms
logx - logy = log ([tex]\frac{x}{y}[/tex] )
logx = logy , then x = y
Given
log(x) - log(x + 6)
= log([tex]\frac{x}{x+6}[/tex] )
Then
log )[tex]\frac{x}{x+6}[/tex] ) = log([tex]\frac{1}{2}[/tex] ) , thus
[tex]\frac{x}{x+6}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
2x = x + 6 ( subtract x from both sides )
x = 6
Rewrite in simplest terms: -8(9r – 1) – 9(-85 + 2)
The Answer is:
−72 r + 755
Answer:
-72r + 755
Step-by-step explanation:
-8(9r – 1) – 9(-85 + 2)
-72r + 8 - 9(-83)
-72r + 8 + 747
-72r + 755
Yoko received a $90 gift card for a coffee store. She used it in buying some coffee that cost $8.23 per pound. After buying the coffee, she had $40.62 left on her card. How many pounds of coffee did she buy?
Answer: 6
Step-by-step explanation:
Step 1: Set up equation:
90=40.62+8.23p
^ the 'p' represents pounds.
Step 2: Subtract 40.62 from both sides
49.38 = 8.23p
Step 3: Divide both sides by 8.23 (We do this to get the variable "p" by itself)
6=p
Trust Milky. Milky Smart
HELPP I WILL GIVE YIU BRAINLIEST!! i think the answer is 42x + 12 but not entirely sure!!!
Answer:
yes your answer is correct
Step-by-step explanation:
there is a line that includes the point 7,2 and has a slope of -1/3
Answer:
y= -1/3x + 13/3
Step-by-step explanation:
we can find the equation of the line thanks to this formula
y-y1 = m(x-x1) where y1 and x1 indicate the coordinates of the point, while m indicates the slope
y-2 = -1/3(x-7)
y = -1/3x+7/3+2
y= -1/3x + (7+6)/3
y = -1/3x+13/3
A point P(x, y) moves along the graph of the equation y = x3 + x2 + 6. The x-values are changing at the rate of 2 units per second. How fast are the y-values changing (in units per second) at the point Q(1, 8)?
The y-values are changing at a rate of 24 units per second at the point Q(1, 8).
To find how fast the y-values are changing at the point Q(1, 8), we need to calculate the derivative of the given equation with respect to x. The derivative of y = x^3 + x^2 + 6 is dy/dx = 3x^2 + 2x.
At the point Q(1, 8), we substitute x = 1 into the derivative to find the rate of change.
dy/dx = 3(1)^2 + 2(1) = 3 + 2 = 5.
Therefore, the y-values are changing at a rate of 5 units per second at the point Q(1, 8). However, the rate at which the x-values are changing is given as 2 units per second.
Thus, to determine how fast the y-values are changing with respect to time, we multiply the rate of change in y by the rate of change in x: 5 units/second * 2 units/second = 10 units/second. Hence, the y-values are changing at a rate of 10 units per second at the point Q(1, 8).
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one card is selected at random from a deck of cards. determine the probability of selecting a card that is less than 8 or a club. note that the ace is considered a low card.
To determine the probability of selecting a card that is less than 8 or a club from a standard deck of cards, we need to consider the number of favorable outcomes and the total number of possible outcomes.
First, let's calculate the number of cards that are less than 8. There are four suits (hearts, diamonds, clubs, and spades), and each suit has cards numbered 2 through 7. So, there are 4 suits * 6 cards per suit = 24 cards that are less than 8.
Next, let's calculate the number of clubs in the deck. There are 13 cards in each suit, and one of those suits is clubs. Therefore, there are 13 clubs in the deck.
To find the probability, we add the number of favorable outcomes (cards less than 8 or clubs) and divide it by the total number of possible outcomes (52 cards in a deck).
Probability = (24 + 13) / 52 = 37 / 52 ≈ 0.7115
Therefore, the probability of selecting a card that is less than 8 or a club is approximately 0.7115 or 71.15%.
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Test the claim about the population mean, h, at the given level of significance using the given sample statistics. Claim: h = 30; « = 0.08; 3 = 3.31. Sample statistics: x = 28.9, n = 65 Calculate the standardized test statistic. The standardized test statistic is -2.68 (Round to two decimal places as needed.) Determine the critical value
The standardized test statistic is -2.68 and the critical value is 1.44.
Given that, Claim: h = 30 Sample statistics: x = 28.9, n = 65 α = 0.08 σ = 3.31
We need to calculate the standardized test statistic and critical value.
Test statistic: The standardized test statistic can be calculated as follows:
Z = (x - μ) / (σ / √n)
Substitute the given values in the above formula to get
Z = (28.9 - 30) / (3.31 / √65) = -2.68
Critical value: Since the level of significance α is given, we can calculate the critical value using the z-score table for standard normal distribution.
Since α is given, the right-tailed test is conducted.
The critical value is obtained by subtracting the α-level from 1 and then find the z-score value.
Let's find the z-score value corresponding to the area 0.92 in the z-score table.
We get the corresponding z-score as 1.44, which is the critical value.
Hence, the standardized test statistic is -2.68 (Round to two decimal places as needed.) and the critical value is 1.44.
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Solve for x
HELP ME PLEASEEEE
Answer:
x = 9
Step-by-step explanation:
5x + 7 = 52
5x = 52-7
5x = 45
x= 9
HELP ASAP!!! find the lines of symmetry. Select all that apply
Answer:
B. and D.
Step-by-step explanation:
If you folded that shape along the lines o and m, it would match up.
There are two lines of symmetry m and o. Options B and D are correct.
From the given figure we have to determine the axes of symmetries.
A line is a straight curve connecting two points or more showing the shortest distance between the initial and final points.
Here,
As of the given figure,
There are 2 axes of symmetry first line m and the other is line o.
Line l and line n are not symmetrical lines because the reflection of one portion will not be superimposed with the other.
Thus, there are two lines of symmetry m and o. Options B and D are correct.
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Suppose that instead of the model inc; = Be + B, ed; + B2 sex, + &i (sex equals 1 for males and 0 for females) we posited inc; = 20 + a, ed, + a2 sexi + ei = 365 where sex* equals 0 for males and 1 for females. What would you expect concerning the relationship between the parameter estimates and the interpretation of these estimates from OLS regressions of these alternative formulations?
The two alternative formulations of the OLS regressions have differences in their parameter estimates and interpretations. The first model can be represented as: inci = Be + B1edi + B2sexi + εi, where sexi equals 1 for males and 0 for females. The second model is given by: inci = a0 + a1edi + a2sexi + ei, where sexi equals 0 for males and 1 for females.
The first model, the intercept term B0 represents the average income earned by men (when sexi equals zero) who have no education. The coefficient B1 represents the increase in income for each additional year of education. B2 represents the average difference in income between men and women when education is the same. Thus, B2 is the coefficient of the dummy variable for sex.In the second model, a0 represents the average income earned by women (when sexi equals one) who have no education. a1 represents the increase in income for each additional year of education, and a2 represents the average difference in income between men and women when education is the same. Thus, a2 is the coefficient of the dummy variable for sex.The interpretation of the coefficients is different between the two models. In the first model, B2 represents the average difference in income between men and women. In the second model, a2 represents the average difference in income between women and men. Thus, the coefficients of the dummy variable for sex have opposite signs and different interpretations. Also, the intercept terms in the two models are different, which reflects the difference in the average income earned by men and women in the two models. The parameter estimates from the OLS regressions for these alternative formulations will differ as well.
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Which of the following is a sine of a in the right triangle below
Answer:
[tex]\frac{5}{13}[/tex]
Step-by-step explanation:
sine of A = [tex]\frac{opposite}{hypotenuse}[/tex]
opposite = 5
adjacent = 12
hypotenuse = 13
Answer:
C.) 12/13
Explanation:
trust me
d = 18 in. Fine the radius or diameter of each circle with the given dimensions.
Answer:
r= 9
Step-by-step explanation:
radius (r) = diameter (D)/2
= 18/2
= 9
(Time complexity, 20pt) For the following languages, sketch (high-level, no need to mention states) a 1-tape TM program that solves the problem and include a time analysis of your program showing how much time it takes, e.g. O(n),O(nlog n), O(n^2), O(n^3), etc.; n is the length of the input.
a) [10pt] L = {u#v: U, v E {0,1}* and u is a substring of v}
b) [CSC 389, 10pt] L = {0^2^k : for k >= 0}. E.g. 0,00,0000,00000000 € L. Note that n = 2^k.
a) Language L = {u#v: u, v ∈ {0,1}* and u is a substring of v}
b) Language L = {0^2^k : for k ≥ 0}
Time Analysis: The time complexity of this program is O(n), where n is the length of the input. This is because we only need to scan through the input string once, counting the consecutive '0' symbols, and check if the count is a power of 2.
a) Language L = {u#v: u, v ∈ {0,1}* and u is a substring of v}
To sketch a 1-tape Turing machine program for this language, we can follow these steps:
1. Start at the leftmost symbol of the input.
2. Scan to the right until we encounter the '#' symbol, marking all the scanned symbols.
3. Once we encounter the '#' symbol, start scanning to the right again, comparing each scanned symbol with the marked symbols.
4. If a match is found, continue scanning until the end of the input, ensuring that the remaining symbols match the unmarked symbols.
5. If a mismatch is found at any point, reject the input.
6. If the end of the input is reached and all symbols match, accept the input.
Time Analysis: The time complexity of this program is O(n^2), where n is the length of the input. This is because, in the worst case, we need to compare each symbol in the input string with the marked symbols, and we may need to scan through the input string twice.
b) Language L = {0^2^k : for k ≥ 0}
To sketch a 1-tape Turing machine program for this language, we can follow these steps:
1. Start at the leftmost symbol of the input.
2. Scan to the right, counting the number of consecutive '0' symbols until a different symbol is encountered.
3. If the count of '0' symbols is not a power of 2, reject the input.
4. If a different symbol is encountered, check if it is the end of the input. If it is, accept the input.
5. If a different symbol is encountered and it is not the end of the input, reject the input.
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Plsssss helps 50 points
Answer:
2000.2 cm³
Step-by-step explanation:
Finding the volume of a cylinder is quite simple if you under what pi (3.14) is. You can watch khan academy. It has some useful videos on solving for volume. It also explains very well. Hope this helps!
A dump truck brought 1⁄3 of a ton of rock on the first trip, 1⁄2 of a ton on the second trip, but on the third trip, had to take back 4⁄5 of a ton. What was the total weight of the rock left at the construction site?
Answer:
1/30 ton of rock left
Step-by-step explanation:
1/3+ 1/2= 5/6
5/6 - 4/5= 1/30 ton of rock left at the construction site.
Pls help me I don’t know how to do it!
Answer:
864
Step-by-step explanation:
To find the area of a shape, you have to multiply length times width.
Answer:
72 ft. sq.
Step-by-step explanation:
The way to look at it it to multiply length times width ( l x w )
there are 2 obvious numbers to get the answer, which is 8 and 9 so 8 x 9 = 72
Let f : R → R be a function. Define a sequence fn : R → R by 1 fn(x) = f ( x + = = (₁ + ₁/ ). n (a) Prove that if f is uniformly continuous, then fn → f uniformly. (b) Prove that uniform continuity of f is a necessary assumption to conclude that fn → ƒ uniformly. That is, give an example of a function ƒ : R → R such that f is continuous, but that fn does not converge to f uniformly.
(a) If f is uniformly continuous, then fn converges to f uniformly.
(b) Uniform continuity of f is necessary for the uniform convergence of fn to f; a counterexample is provided with a continuous f but non-uniform convergence of fn.
(a)If f is uniformly continuous, then we want to show that fn converges to f uniformly.
Given ε > 0, since f is uniformly continuous, there exists δ > 0 such that for any x, y in R, if |x - y| < δ, then |f(x) - f(y)| < ε.
Now, consider fn(x) = f(x + 1/n) for each n. We need to show that for any ε > 0, there exists N such that for all n ≥ N, |fn(x) - f(x)| < ε for all x in R.
Fix ε > 0. By the uniform continuity of f, there exists δ > 0 such that for any x, y in R, if |x - y| < δ, then |f(x) - f(y)| < ε.
Now, for any x in R, let N > 1/δ. Then for all n ≥ N and for any y = x + 1/n, we have |x - y| = |x - (x + 1/n)| = 1/n < δ.
By choosing y = x + 1/n, we have |fn(x) - f(x)| = |f(x + 1/n) - f(x)| < ε.
Since ε was chosen arbitrarily, this shows that fn converges to f uniformly.
(b) Counterexample: To show that uniform continuity of f is necessary for the uniform convergence of fn to f, we need to find an example where f is continuous but fn does not converge to f uniformly.
Consider the function f(x) = x on R. This function is continuous.
Now, let's examine the sequence fn(x) = f(x + 1/n). For any x in R, we have:
|fn(x) - f(x)| = |(x + 1/n) - x| = 1/n.
Notice that for any fixed x, as n approaches infinity, 1/n approaches 0. However, the convergence is not uniform since the choice of x affects the convergence rate.
For any ε > 0, no matter how large N is chosen, there will always exist x such that |fn(x) - f(x)| = 1/n ≥ ε, for n > 1/ε.
Therefore, fn does not converge uniformly to f(x) = x, despite f(x) being continuous.
This counterexample demonstrates that the assumption of uniform continuity of f is necessary for the uniform convergence of fn to f.
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12 inches in this circle represents its
Answer:
I can't see the whole picture, but if it's half of the circle, it's the radius. if the line goes all the way through the circle, it's the diameter
Answer:
I think it’s the diameter
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